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Let A and B be two finite sets. Let f: A -->B be any map. Assume that |A|=|B|.
(i) Show that if f is surjective, then f is injective.
(ii) Show that if f is injective, then f is surjective.
Solution
i)
Let f be surjective.
Let f not be injective
Let, a and b be two distinct elemetns in A so that
f(a)=f(b)
IN a map one element can map to only one element
Hence, remaining |A|-2 elements other than a and b map to at most |A|-2=|B|-2 elements in B
a and b map to f(a)
Hence , f maps to at most |B|-2+1 =|B|-1 elements in B
But f is sujrective to |f(A)|=|B|
HEnce contradiction
So, f is injective
ii)
Let, f be injective
Let, |A|=|B|=n
Let, elemetns of A as
a1,..,an such ai is not equal to aj with i is not equal to j
and elements of B as
b1,b2,...bn so tha
f(ai)=bi
f is injective so
f(ai) is not equal to f(aj) for i not equal to j
Hence, b1,...bn are are mutually distinct
Hence, f(A) has n=|B| elements
Hence, f is surjective

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